How can I melt an object and displace it horizontally?
As a test to find the correct wax melting settings, setup the following.


Open blender and keep the default cube, will be the fluid domain
Place another cube inside it
Resize it to about half of the first cube and press CTRL+A for Apply and select Scale
Enter into Edit mode and subdivide the small cube about 50 times or more.
Go to Physics and select Fluid and the Type to Fluid 
Select the bigger cube now and at the Physics tab select Fluid and the Type to be Domain 
Scroll down to the Fluid World and set it up the way you see it in the image below, Time End and Speed settings also have an effect





Click on the bake, Once it finished you can play the animation and
see the result. Do not forget to hide from Viewport and Rendering the cube that      you setup to be the wax. Clicking the eye and the camera next to the wax object, see image top Cube.WaxCube object.


Experiment with those settings to find the effect you are looking for, and after every change bake it to get the new settings to take effect. If you reduce the Compressibility further, would not go below 0.001, your bake time increases and if you increase the Grid Levels then the same will happen.
Once you find the right settings then you can implement it in your project with materials and the right resolution and so on.
The quality will be depending on the Final Resolution settings which also will effect your bake time, but for final I would set it to 200 at least to make it look good.

For you specific needs you need to create objects that are set up within the fluid domain and they need to have a Physics assigned as Fluid and the Type to be Obstacle. So your wax fluid cube need to be placed into a cube with a hole on it, if you want it to see the wax level drop and flow out. Another object should 'catch' that liquid, again as an obstacle object and the wax fluid will take its shape simulating its solidification. Need to make sure that the object is open on the top otherwise the wax will run on its surfaces.
An example below



Do not forget to set the Physics for the wax holding container that has the hole in it to allow the side flow, like below with the Shell selected. 



Also ad a Subdivision Surface modifier and set it to View and Render to 4.

The below was created with the same setup but removed the cylinder and moved the fluid domain up to the bottom of the wax cube and assigned a material to it to make it visible. The wax 'catcher' in this case is only the fluid domain. The hole on the wax holding cube allowed the flowing out of it side ways. Of course, it was baked again for the new positions. This is only a rough demonstration.